# Expected value problem, 10 floors and 12 people get on an elevator

A building has 10 floors above the basement. If 12 people get into an elevator at the basement, and each choses a floor at random to get out, independently of the others, at how many floors do you expect the elevator to make a stop to let out one or more of these 12 people??

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HINT: For $k=1,\dots,10$ let $X_k$ be a random variable that takes the value $1$ if at least one person gets off on floor $k$ and $0$ otherwise. The probability that no one gets off on floor $k$ is $\left(\frac9{10}\right)^{12}$, so $$\Bbb E(X_k)=1-\left(\frac9{10}\right)^{12}\;.$$ You want $\Bbb E(X_1+X_2+\ldots+X_{10})$, and expectation is linear, so ... ?